OPT  S7 

The  Unilateral  Dynamic  Characteristics 
of  Three-Electrode    Vacuum  Tubes. 


BY 


JOHN  G.  FRAYNE 


A  THESIS 

SUBMITTED  TO  THE  FACULTY  OF  THE  GRADUATE  SCHOOL  OF  THE 

UNIVERSITY  OF  MINNESOTA  IN  PARTIAL  FULFILLMENT  OF 

THE  REQUIREMENTS  FOR  THE  DEGREE  OF  DOCTOR 

OF  PHILOSOPHY. 


Reprinted  from  the  PHYSICAL  REVIEW,  N.  S.,  Vol.  XIX.,  No.  6,  June,  1922. 


The  Unilateral  Dynamic  Characteristics 
of  Three-Electrode    Vacuum  Tubes. 


BY 

JOHN  G.  FRAYNE 


A  THESIS 

SUBMITTED  TO  THE  FACULTY  OF  THE  GRADUATE  SCHOOL  OF  THE 

UNIVERSITY  OF  MINNESOTA  IN  PARTIAL  FULFILLMENT  OF 

THE  REQUIREMENTS  FOR  THE  DEGREE  OF  DOCTOR 

OF  PHILOSOPHY. 


Reprinted  from  the  PHYSICAL  REVIEW,  N.  S.,  Vol.  XIX.,  No.  6,  June,  1922. 


Q 


[Reprinted  from  THE  PHYSICAL  REVIEW,  N.S.,  Vol.  XIX.,  No.  6,  June,  1922.] 


THE    UNILATERAL    DYNAMIC    CHARACTERISTICS    OF 
THREE-ELECTRODE  VACUUM   TUBES.1 

BY  JOHN  G.  FRAYNE. 

SYNOPSIS. 

Unilateral  dynamic  characteristics  of  vacuum  tube  when  plate  circuit  includes 
resistance,  inductance  or  capacity. — (i)  Theoretical  equations.  For  the  case  of  pure 
resistance  (R),  the  Van  der  Bijl  parabolic  relation  between  plate  current  and  effec- 
tive grid  voltage  is  expressed  as  a  power  series  in  e  sin  pt,  the  impressed  simple  har- 
monic grid  voltage.  The  coefficients  of  the  various  harmonics  involve  R,  the  nth 
harmonic  reaching  a  maximum  value  when  R  equals  (n-2)  /3  times  Ro  the  tube  re- 
sistance. For  the  fundamental  the  maximum  energy  output  for  a  given  plate  bat- 
tery is  secured  when  R  —  0.8 1  l?o.  The  dynamic  characteristic  was  obtained  by 
compounding  the  harmonics  into  a  single  curve;  it  approaches  a  straight  line  as  the 
resistance  is  increased.  For  the  case  of  pure  inductance,  the  plate  current  is  ex- 
pressed as  a  Fourier  series.  The  dynamic  characteristic  is  a  closed  loop  whose  area  is 
proportional  to  the  energy  in  the  inductance.  This  loop  reduces  to  an  ellipse  for 
small  values  of  e,  in  which  case  the  tube  functions  as  an  alternator  \vhose  internal 
impedance  is  a  function  of  the  external  load.  The  insertion  of  a  condenser  instead  of 
an  equivalent  inductance  gives  identical  results  except  that  the  phase  angle  of  the 
various  harmonics  is  shifted.  (2)  Experimental  verification.  The  effects  on  the 
plate  current  of  varying  the  alternating  grid  voltage  e,  the  static  grid  voltage  Ec 
and  the  plate  voltage  Et,,  for  a  given  value  of  resistance  R  or  inductance  I,  and  the 
effect  of  varying  R  or  I  with  constant  Eb,  Ec  and  e  (15  or  20  volts),  were  determined 
and  are  shown  in  curves  together  with  the  corresponding  theoretical  values.  A.  W.  E. 
205B  tube  was  used.  The  results  show  that  the  equations  predict  the  harmonic 
constituents  of  the  plate  current  as  high  as  the  fourth,  for  values  of  e  up  to  15  or  20 
volts  (depending  on  £j),  the  range  for  which  the  fundamental  equation  holds.  For 
this  range  the  coefficients  of  the  various  harmonics  in  the  equation  are  proportional 
simply  to  en.  The  fundamental  becomes  greater  while  the  other  harmonics  di- 
minish as  we  approach  the  straight  portion  of  the  static  characteristic  and  as  we  in- 
crease the  plate  potential. 

Circuit  for  producing  pure  sine  wave  electromotive  force  with  frequency  of  200,000 
cycles. — The  oscillating  circuit  and  filters  used  are  shown  diagrammatically  in  Fig.  i. 

Pure  resistance  for  high  frequencies. — A  platinized[quartz  fiber  (diameter  o.oi  mm.) 
with  a  resistance  of  100  ohms  per  inch  will  carry  0.06  ampere  when  immersed  in  acid- 
free  paraffin  oil  and  has  a  negligible  skin  effect. 

f  T  is  a  well-known  fact  that  the  current  flowing  from  a  hot  filament  to 
*•  the  plate  of  a  three-electrode  vacuum  tube  does  not  vary  as  the  first 
power  of  the  plate  potential.  With  a  view  to  determining  what  this 
relation  really  was,  theoretical  and  experimental  investigations  were 
undertaken  by  Langmuir,2  Bethenod,3  Vallauri,4  Van  der  Bijl,5  Latour6 

1  Presented  at  the  Chicago  meeting  of  the  American  Physical  Society,  December,  1920. 

*  P.  I.  R.  E.,  3,  261-93,  Sept.,  1915,  and  PHYS.  REV.,  2,  p.  457,  1913. 

1  La.  Lum.     EL,  35,  25-31,  Oct.  14,  1916. 

4  L'Elettrotecnica,  Vol.  4,  Nos.  3,  4,  18  and  19,  1917. 

6  PHYS.  REV.,  n,  p.  172-198,  1918. 

6  La  Lum.     El.,  Dec.  30,  1916. 


• »  '  «  '-'JOHN   G.   FRAYNE. 

and  others.  The  second  degree  equation  obtained  by  Van  der  Bijl  lends 
itself  more  easily  to  mathematical  treatment  than  any  of  the  others, 
and  agrees  very  closely  with  experimental  evidence  over  a  certain  range 
of  plate  and  grid  potentials. 

The  curves  obtained  by  plotting  the  plate  current  against  the  grid 
voltage  for  given  plate  potentials  are  usually  referred  to  as  the  static 
characteristics  of  the  tube.  The  term  "dynamic  characteristic"  is  used 
when  the  grid  potential  is  of  an  oscillating  nature.  The  latter  charac- 
teristic is  usually  referred  to  as  being  "unilateral"  when  there  is  no 
external  coupling  between  the  grid  and  plate  circuits,  as  distinguished 
from  "regenerative"  when  such  coupling  exists.  Van  der  Bijl  has  shown 
that  the  insertion  of  a  resistance  between  the  plate  and  the  plate  battery 
changes  the  form  of  the  dynamic  characteristic  from  a  parabola  to  a 
curve  which  approaches  a  straight  line  with  increasing  resistance.  A 
solution  similar  to  that  of  Van  der  Bijl  is  obtained  here,  and  in  addition 
the  case  where  the  resistance  is  replaced  by  an  inductance  is  worked  out. 
We  shall  consider  three  cases  here.  First,  with  no  resistance  in  the  plate 
circuit,  secondly,  with  a  resistance  inserted,  and  finally  with  the  latter 
replaced  by  an  inductance. 

CASE  OF  No  EXTERNAL  RESISTANCE. 
Let  Eb  —  plate  potential, 
/&    =  plate  current, 
EC  =  grid  potential, 

e  sin  pt  =    superimposed  e.m.f.  on  grid. 
According  to  the  current-squared  law 

Ib  =  A(Eb  +  nEc  +  ne  sin  pt  +  e)2,  (i) 

where  /*  is  the  amplification  constant,  defined  by 


and  A  and  e  are  constants  depending  on  the  structure  of  the  tube.  In  this 
case  the  grid  potential  has  the  value  Ec  +  e  sin  pt.  The  equivalent  plate 
potential  is  therefore  /x  (Ee  +  e  sin  pt). 

Before  proceeding  further  it  might  be  well  to  remark  here  that  in  order 
that  (i)  may  actually  represent  the  true  conditions  the  value  of  e  must 
lie  within  certain  limits,  namely 

77,   -L  * 

Ec 


e  =i  \EC 


e  s 


A* 

where  g  is  the  maximum  positive  voltage  the  grid  can  have  before  it 


VoL.^XIX.J  THREE-ELECTRODE    VACUUM   TUBES.  63! 

begins  to  attract  many  electrons.  Also  if  e  sin  pt  attains  such  a  large 
negative  value  in  the  cycle  that  the  expression  above  is  negative,  the  re- 
sulting current  wave  will  be  flattened  out  at  that  part  of  the  character- 
istic curve.  Equation  (i)  might  be  written  generally  as: 

Ib  =  f  (fj.e  sin  pt)  . 
Expanding  by  Maclaurin's  Theorem 


/•-  -  /  (o)  +  W  sin  ptf  (o)  +  Ptf"  (o) 


and  since  Ib  is  the  same  function  of  Eb  as  it  is  of  sin  pt 

=2A  ^  +  M£C  +  C 


=  2  A. 

\  d  Eb2  Jt=o 
Therefore 

.   u.e  sin  pt      A/J?  ez  .  Ap?  ez     ,     , 

Ib  =  A  (Eb  +  fM  Ec  +  c)2  +    — — j " —  cos  2pt  +  •-&—  .  (20) 

Thus  in  the  simple  case  illustrated  above  where  the  plate  potential  is 
kept  constant  throughout  the  operation,  a  pure  sine  wave  on  the  grid  gives 
rise  to  a  current  of  the  same  frequency  (called  the  fundamental)  in  the 
plate  circuit,  a  first  harmonic  and  a  rectified  current  component.  In 
this  case  the  actual  dynamic  and  static  characteristic  curves  will  coincide. 

CASE  OF  A  PURE  RESISTANCE. 

Next  we  shall  consider  the  case  where  the  potentials  on  the  grid  and 
plate  vary  simultaneously.  Let  a  resistance  R  be  connected  between 
the  plate  and  the  plate  battery.  Then 

Ib  =  A  {E  -  R  Ib  +  M  (Ec  +  e  sin  pi)  +  e}2.  (3) 

This  can  be  expanded  as  an  infinite  series. 

J»=  /(O)     |    /'(o)MgGin^+/"(°) 

2! 
(tie  sin  pi)2  +  •   •   •        °   (pe  sin  pi)n  (4) 

Van  der  Bijl  has  shown  that  since  the  parabolic  relation  connecting 
plate  current  and  grid  and  plate  potentials  is  only  an  empirical  approxi- 
mation, it  is  not  to  be  expected  that  the  higher  derivatives  in  the  series  will 
accurately  represent  the  actual  experimental  values.  However,  the 


632 


JOHN   G.   FRAYNE 


[SECOND 

LSEklES. 


derivatives  up  to  probably  the  third  or  fourth  ought  to  be  a  close  approxi- 
mation and  the  higher  derivatives  ought  to  indicate,  at  least  in  a  quali- 
tative way,  how  the  higher  harmonics  depend  on  the  various  tube  con- 
stants and  on  the  properties  of  the  external  circuits. 

Referring  back  to  equation  (4),  the  coefficients  of  the  series  are  as 
follows : 

+  — 

Ro  ' 

where 


/(o)  = 


being  defined  as 


2  A 


/(o)=^    i-B- 


=  _2 


£-5/2 


The  general  functional  term  is  given  by 

fn  (0)    =  ( LL_ ^L. 


The  coefficient  of  sin  (££)  is  therefore  the  value  of  this  expression  mul- 
tiplied by  (ve)n.     If  this  coefficient  is  denoted  by  «„,  then 


Limit 

n  =  «. 


«n  +   I 


2  («  +  2) 


(*  +  I)  S 


In  order  that  the  series  (4)  may  be  absolutely  convergent 


2RA 
~B~ 


<  i. 


Therefore 


e  < 


B 


2RA  /*' 


Thus  for  a  given  A,  n  and  J?,  the  smaller  the  value  of  R0,  the  greater  e 
may  be. 

Using  the  values  of  A,  AC,  R  and  J?o  given  later,  e  may  have  values 
reaching  up  to  150  volts.  However,  it  will  be  seen  later  that  in  practice 
e  cannot  have  a  value  larger  than  about  15  volts  if  equation  (3)  is  to 
represent  conditions  accurately.  The  physical  limitations  which  the 
tube  imposes  on  the  characteristic  equation  make  it  impossible  to  use 


VoL.^XIX.J  THREE-ELECTRODE    VACUUM   TUBES.  633 

grid  voltages  more  than  one  tenth  of  the  limiting  value  as  given  by  (5). 
It  is  very  evident  that  for  small  values  of  e,  the  series  (4)  converges 
rapidly  and  in  consequence  only  a  few  terms  need  be  evaluated  in  order 
to  find  a  close  approximation  to  the  actual  current  flowing  under  a  cer- 
tain condition  of  the  amplifier. 

Now  /'  (0)  stands  for  the  reciprocal  of  the  total  output  resistance  R'0 
when  there  is  an  external  resistance  in  the  plate  circuit. 

Therefore 

i         i 


The  total  resistance  of  the  complete  plate  circuit  is  thus  a  rather  compli- 
cated function  of  the  external  resistance  and  the  internal  output  resist- 
ance of  the  tube  when  there  was  no  resistance  in  the  plate  circuit. 

Since  the  series  (4)  is  a  power  series  in  sin  (pi)  it  is  necessary  to  con- 
vert the  various  powers  of  sin  (pi)  into  first-power  expressions  of  func- 
tions of  multiples  of  pt,  and  expressions  corresponding  to  the  rectified 
currents.  Since  the  series  converges  rapidly  for  values  of  input  voltage 
within  the  limits  (2),  all  powers  of  sin  (pt}  beyond  the  fourth  will  be 
omitted. 


+  1  I  ^  (i  -  B~1'2)  1  -  ^  AZR  M3  e3  £~5/2l  sin  (#) 

[I  s  1 

-A  v?e2  B~W  +  -A*R?n*e*  B~7/2     cos  (2  pt  +  TT) 
2  2  J 

+  -  \A*  R  M3  e3  5~5/2 1  sin  3  pt 
I  ^3  7?2  /i4  e4  £-7/2 1  cos  4  pt  + 


Actual  computation  of  the  coefficients  in  this  series  show  that  for  values 
of  e  within  the  limits  specified  above,  the  series  converges  very  rapidly. 

For  values  of  e  below  10  volts,  actual  computations  show  that  the  co- 
efficient of  sin  (pt}  is  practically  a  linear  function  of  e;  beyond  ten  volts 
the  term  involving  e  becomes  appreciable  and  the  relation  becomes  more 
complex.  Similarly  the  coefficient  of  cos  (2  pt  +  TT)  varies  as  the  square 
of  e  up  to  about  i  o  volts.  Since  we  have  taken  no  powers  higher  than 
sin  pt  the  coefficients  of  the  third  and  fourth  harmonics  vary  directly  as 
the  cube  and  fourth  powers  respectively  of  e. 

The  relation  between  the  coefficients  and  R,  when  the  latter  is  variable, 
can  best  be  shown  by  examining  the  condition  for  maxima.  If  we  take 


634  JOHN  G. 

the  wth  derivative  as  representing  the  coefficient  of  the  general  term, 
then  the  latter  will  be  a  maximum  when 

d 

dR 

i.e.,  when 

"  R0     or    (i  +  2  R/R0)-1  =  o. 

The  latter  equation  has  a  solution,  R  =  <x>  .  This  is,  obviously,  the  con- 
dition for  a  minimum.  The  first  relation  shows  that  R  must  be  a  nega- 
tive quantity  for  n  =  I.  Hence  the  fundamental  has  no  real  maximum. 
For  n  =  2,  the  maximum  occurs  at  R  =  o.  For  n  —  3,  the  maximum  occurs 
when  the  external  resistance  is  one  third  of  the  tube  resistance.  For 
the  higher  derivatives,  the  position  of  the  maxima  occur  at  continuously 
increasing  values  of  R. 

If  the  amplitude  of  the  impressed  e.m.f.  is  less  than  15  volts,  equation 
(3)  holds  good  for  all  values  of  R,  when  the  plate  voltage  is  maintained 
at  200  volts,  and  the  grid  voltage  is  —7.5.  For  values  of  e  over  15  volts, 
using  the  same  grid  and  the  plate  potentials,  equation  (3)  no  longer  holds. 
Hence  the  amplitudes  of  the  harmonics  as  experimentally  found  for 
values  of  e  over  15  volts  depend  on  other  features  of  the  amplifier.  Since 
Ec  is  —  7.5  the  grid  will  be  raised  to  a  positive  potential  of  7.5  volts 
during  this  cycle.  In  Fig.  5  it  will  be  noticed  that  at  this  value  of  Ec 
on  the  2oo-volt  parameter,  the  static  characteristic  begins  to  lose  its 
parabolic  nature  and  tends  to  flatten  out.  From  the  nature  of  the  static 
characteristic  it  may  be  seen  that  the  higher  the  plate  voltage  is  raised 
the  greater  the  values  e  may  have  and  remain  within  the  proper  limits. 
This  amounts  to  saying  that  the  smaller  R0  is,  the  greater  the  input  volt- 
age on  the  grid  may  be. 

The  dynamic  characteristics  for  this  case  may  be  obtained  as  follows: 
The  instantaneous  values  of  the  various  harmonics  for  values  of  pt  be- 
tween o  and  27T  are  plotted,  and  then  these  constituent  sine  waves  com- 
pounded to  give  the  actual  wave  shape.  If  now  the  resulting  periodic 
current  is  plotted  along  the  /&  axis  and  the  input  voltage  plotted  on  the 
EC  axis,  the  resulting  curve  will  be  the  so-called  dynamic  characteristic 
of  the  tube  under  the  specific  conditions.  It  will  be  seen  that  if  all  terms 
but  the  fundamental  had  been  neglected,  the  characteristic  would  have 
been  a  straight  line.  Addition,  however,  of  the  first  harmonic  causes 
the  characteristic  to  have  a  definite  curvature.  The  smaller  the  value 
of  the  external  resistance,  the  more  nearly  does  the  curve  approach  the 
parabolic  relation  holding  in  the  static  case,  and,  of  course,  in  the  limiting 
case  when  R  is  zero,  the  two  characteristics  coincide. 


Na6XIX']  THREE-ELECTRODE  VACUUM  TUBES.  635 

CONDITION  FOR  MAXIMUM  OUTPUT. 

In  connection  with  the  expression  for  the  internal  resistance,  it  may 
be  pointed  out  that  the  usual  statement  that  the  maximum  power  is 
obtained  from  a  tube  when  the  external  resistance  in  the  plate  circuit  is 
equal  to  the  internal  output  resistance  of  the  tube  needs  clarification. 
If  by  maximum  power  is  meant  the  greatest  power  obtained  from  the 
fundamental  frequency,  the  following  is  valid. 


Power  =  RP  =  ^  \  i  —  B~  (6) 

Therefore 

dP  _     _  /A 
dR  "          jR 

<fP       ^r 

—  =  G>  tor  maximum  r. 

dR 

Therefore 

.  =  i  ± 


£3/2  =  2B  _  !    or 


> 
RO          4 

The  condition  for  a  maximum  dissipation  of  fundamental  current 
energy  is  that  the  ratio  of  R  to  the  internal  resistance  when  R  was  zero 
is  .81,  This  condition  holds  in  the  case  where  the  maximum  power  is 
desired  with  a  certain  fixed-plate  battery,  and  a  variable  resistance  is 
available.  The  usual  condition  for  maximum  power,  that  the  internal 
and  external  resistances  be  equal,  is  only  true  in  this  case  if  the  actual 
plate  potential  is  kept  constant  while  the  plate  resistance  is  varied.  The 
condition  under  which  the  above  relation  was  obtained  is  the  one  most 
commonly  met  with  in  practice. 

CASE  OF  AN  INDUCTANCE. 

When  an  inductance,  /,  is  placed  between  the  plate  and  plate  battery, 
the  equation  for  the  plate  current  may  be  written  as  follows: 

7  =  A  \Eb  -  l~    +  p  (Ec  +  e  cos  p£)  +  e  f         or         (9) 


B2  -  2  B  L~  -  2LF  cos  pt  ~  +  2  BFcos  pt  +  L2  (~\+  F2  cos2  pt 
at  dt  \dt  ) 

where 

B  =  A1'2  (Eb  +  vZc  +  «)•    L  =  A1'2,     F  =  Al'*ne.  (10) 

A  rigorous  solution  of  this  differential  equation  for  7  is  very  difficult. 
but  an  approximate  method  of  solving  it  may  be  legitimately  utilized. 
Experimental  evidence  shows  that  /  is  a  rapidly  converging  Fourier 


636  JOHN   G.   FRAYNE 

series,  and  that  the  frequency  of  the  fundamental  is  the  same  as  the  fre- 
quency of  the  input  e.m.f.  on  the  grid. 
We  can  therefore  write : 

oc  oc 

J  =  a0/2  -f-  ^  an  sin  npt  +  ^  /3n  cos  npt  (n) 

n=l  7i=l 

In  terms  of  the  exponential  values  for  the  sine  and  cosine, 

2an  sin  npt  +  2/3n  cos  npt  =  (pn  -  ian}  e  inpt  +  (ft,  +  ian)  e  -  inpt. 

Write 

2an  =  pn  —  ian,         and         2bn  =  /3n  +  ictn.  (12) 

Then 

/  =  a0/2  +  f;   an  e  inpt  +^bne  *»i".  (13) 

«-=!  71  =  1 

If  we  substitute  this  value  of  I  in  equation  (10),  we  can  arrange  the 
resulting  terms  in  ascending  orders  of  eipt  and  in  descending  orders  of 
eipt.  The  expression  will  not  be  given  here  as  it  is  very  lengthy  and 
cumbersome.  Since  we  have  terms  involving  elpt  and  elpt  and  corres- 
ponding higher  powers  of  e  on  both  sides  of  the  equation,  the  coefficients 
of  the  like  powers  on  either  side  may  be  equated.  Hence  we  obtain  a 
series  of  2n  equations  from  which  the  a's  and  b's  can  theoretically  be 
determined.  The  general  solution  of  these  equations,  while  ideally  pos- 
sible, is  impracticable  without  further  assumptions  as  to  the  nature  of 
the  coefficients.  We  saw  in  the  case  of  the  resistance  of  the  plate  cir- 
cuit, that  only  the  first  few  terms  of  the  series  were  of  importance  for 
values  of  e  within  the  limits  of  equation  (2),  and  that  for  values  of  e  up 
to  10  or  15  volts,  the  amplitude  of  the  fundamental  varied  approximately 
as  the  first  power  of  e.  If  all  the  coefficients  other  than  a0,  ai,  and  bi, 
are  negligible,  we  have: 

BAWpe 

bi  =  —  - ,  (14) 

i  —  2BLip 

showing  that,  for  this  case,  a\  and  &i  are  linear  functions  of  e.  Substi- 
tuting the  above  values  of  a\  and  61  in  the  expressions  for  az  and  b2,  we 
have : 


4(1  +  (     } 


~  4  (i  -  2  BLipY  (i  -  4  BLip)  ' 

The  values  of  az  and  b%  were  found  on  the  assumption  that  all  the  higher 
coefficients  were  negligible.     Similarly  o3  and  63  may  be  found  and  so  on. 


THREE-ELECTRODE  VACUUM  TUBES.  637 

From  relation  (22)  the  values  of  ai,  /3i,  az,  /32,  etc.,  may  be  found,  and 

fj,  e  cos  (pt  —  a)       .  _.  Ip 

ai  sm  pt  +  ft,  cos  pt  =  . .          ,  where  a  =  tan  l:3£       (16) 

(jKo  + 


and  RQ  = 


e)  ' 


.4ju2e2  cos  (2  pt  -  j8) 
sm  2  *  +  ft  cos  2  *  =  ,       (17) 


where 

/3  - 


Similarly  a3  sin  3  pt  +  /33  cos  3  £/  may  be  found  and  so  on  for  the 
higher  terms. 

Since  I  =  a0/2  +  2an  sin  («£/)  +  2/3w  cos  w^>^,  the  addition  of  the 
various  quantities  found  above  will  give  the  resulting  current  7. 

It  is  obvious  that  as  soon  as  the  values  of  a%  and  b%  become  appreciable 
compared  with  a\  and  bi,  the  values  of  the  latter  obtained  above  can  no 
longer  be  correct,  since  they  were  determined  on  the  basis  that  all  the 
other  coefficients  were  negligible. 

If  the  values  obtained  for  o2  and  b2  are  substituted  in  the  equations 
for  ai  and  bi,  the  following  is  the  value  of 

fj.e  cos  (pt  (pt  —  a) 


where 


2  R05  A*y?e*  Ip  cos  (pt  -  e)  .     . 

(R<?  +  p  p*y  (RQZ  +4  pp)  ilz  ' 


.       _.  ..  ?  -2R02\ 

a  =  tan  :  -    e  =  tan  1  ~   -   and       -  „  )  . 

<?-5PP2J 


2  lp 


In  order  to  obtain  a  numerical  value  for  a\  sin  (pt)  +  jSx  cos  pt,  it  is  best 
to  evaluate  each  term  separately  and  then  compound  the  results  by  the 
parallelogram  law.  Similarly  if  the  values  of  o3  and  b3  become  com- 
parable with  az  and  &2,  we  find  for  the  corrected  value  of 

R03  Atf&  cos  (2  pt  -  |8) 
«2  sin  2  pt  +  focos  2  £/  = 


2       0 

12  RJ  A3  fji4e4  12p2  cos  (2  ^  -  X) 


where  /3  is  the  same  as  defined  in  (27)  and 

3  ^  ~  I7 


X-      an-* 


638  JOHN   G.   FRAYNE. 

By  making  successive  approximations  as  many  terms  as  desired  may 
be  included  in  the  expressions  for  any  particular  harmonic.  It  will  be 
noticed  that  the  resulting  angle  of  lag  of  each  harmonic  depends  on  the 
number  of  terms  we  include  in  the  coefficient,  and  thus  there  arises  a 
peculiarity  in  a  vacuum-tube  generator,  namely  that  the  angles  of  lag  of 
the  various  output  harmonics  are  dependent  on  the  amplitude  of  the 
input  wave  on  the  grid.  The  larger  the  amplitude  for  the  coefficients  of 
the  various  harmonics,  and  the  consequent  shifting  of  the  angles  of  lag 
results. 

AREA  OF  THE  CHARACTERISTIC  LOOP. 

Since  the  fundamental  plate  current  lags  behind  the  grid  voltage  by 
an  angle  a  =  tan"1  (Ip/R)  ,  it  is  evident  that  if  this  current  be  plotted  against 
the  alternating  grid  voltage  an  elliptical  characteristic  will  be  produced. 
If,  however,  the  first  and  higher  harmonics  are  plotted  in  addition  to  the 
fundamental,  and  the  curves  thus  formed  compounded  into  a  single 
curve,  it  is  evident  that  the  characteristic  will  no  longer  be  a  true  ellipse. 
Since  the  amplitudes  of  the  harmonics  are  small  compared  to  that  of  the 
fundamental,  the  resulting  curve  will  not  seriously  depart  from  an  ellipse. 
This  curve  is  what  is  usually  referred  to  as  the  dynamic  characteristic. 
The  a0/2  term  of  the  series  gives  the  point  of  operation  on  the  static 
characteristic,  and  it  is  obvious  from  the  expression  for  the  latter,  that 
the  larger  the  harmonics  become,  the  greater  is  the  shift  of  this  point 
of  operation.  In  practice  this  shift  is  noticed  by  the  increased  reading 
of  a  direct-current  milliameter. 

If  all  but  the  fundamental  current  is  omitted,  the  area  of  the  loop  may 
be  easily  found. 

Put  x  =  Ec  +  e  cos  (pt) 


Limits  for  pt  are  o  and  2ir 

X2T  /»2T      I  „    £  1 

ydx  =  -          {  ao/2  +  cos  (pt  -  a)  [  sin  pt  dt 

Jo     I  (Ro  +  fW  J 

27r  ez  u  sin  a  2ir  /j.  e2  Ip        .  _.  Ip 

—  .    _     —    _  •  _  *•  _  _        ci  t"|  (*f*  sy     —  *     T~O  t"»      -*-  ,-„  _ 

~  (J?o2  +  W2    "    CRo2  +  Pp2)  '  Ro' 

If  the  curve  be  referred  to  the  /&,  Eb  axes,  this  expression  must  be  mul- 
tiplied by  /i.  Also  since  the  maximum  value  70  of  the  fundamental  is 
jue  (J?02  +  I2p2)~1/2,  the  area  of  the  loop  may  be  written  as 

A  =  2  -xlf  lp  =  27r^£0/o2. 
•S-o 


NoL6XIX']  THREE-ELECTRODE    VACUUM    TUBES.  639 

Since  //02/2  =  the  maximum  dynamical  energy  in  the  inductance,  the 
area  of  the  loop  is  thus  proportional  to  that  quantity.  If  the  inductance 
were  not  present  A  =  o,  which  means  that  the  characteristic  no  longer 
has  the  form  of  a  loop,  but  reverts  back  to  the  type  found  when  a  resist- 
ance was  placed  in  the  plate  circuit. 

CASE  OF  A  CAPACITY  IN  THE  PLATE  CIRCUIT. 

Since  a  condenser  placed  between  the  plate  and  the  battery  prevents 
the  direct  current  from  flowing,  it  is  necessary  to  place  a  choke  coil  across 
the  condenser.  The  choke  coil  will  allow  the  direct  current  to  pass, 
but  if  made  properly  will  offer  almost  an  infinite  resistance  to  the  high- 
frequency  current. 

The  solution  for  this  case  is  directly  analogous  to  that  for  the  induct- 
ance problem,  the  only  difference  in  the  final  result  being  that  i/cp 
always  replaces  Ip. 

Thus  the  simple  expression  for  the  fundamental  becomes  iJ.ej(R^ 
+  i/czp2)ll2cos(pt  +  a)  whereon  =  tan"1  i/R0cp.  Similar  expressions  for 
the  other  harmonics  may  be  found  from  comparison  with  the  expressions 
found  for  the  inductance. 

The  dynamic  characteristic  for  this  case  is  similar  to  that  found  for  the 
inductance,  the  only  difference  being  that  it  is  traced  out  in  the  opposite 
direction. 

DESCRIPTION  OF  APPARATUS  AND  EXPERIMENTAL  PROCEDURE. 
In  order  to  have  an  experimental  set-up  which  could  be  used  to  verify 
the  preceding  theory,  the  following  conditions  and  requirements  had  to 
be  met. 

(a)  Production  of  a  pure  sine  wave  e.m.f. 

(b)  Use  of  a  sufficiently  low  frequency  that  capacity  effects  might  be  of 

small  magnitude. 

(c)  Accurate  measurement  of  the  input  e.m.f.  on  the  grid  of  the  har- 

monic producing  tube. 

(d)  Use  of  a  pure  resistance. 

(e)  Use  of  a  pure  inductance. 

(/)    Measurement  of  the  amplitude  of  the  harmonics  produced,  without 
introducing  extraneous  resistances,  etc.,  into  the  harmonic  producer. 

Fig.  i  is  the  complete  circuit  diagram  of  the  entire  collection  of  appar- 
atus used  in  the  experimental  work.  It  may  be  divided  into  three  main 
sections,  the  oscillator,  harmonic  producer  and  the  harmonic  ana- 
lyzer. The  oscillator  in  the  upper  left  corner  is  designed  so  as  to  pro- 
duce as  pure  a  sine  wave  as  possible.  The  tuned  circuit  LlCl  prevents 


640 


JOHN   G.   FRAYNE. 


[SECOND 
[SERIES. 


the  fundamental  frequency  from  passing  into  the  battery  circuit,  thus 
compelling  it  to  travel  to  the  filament  through  the  inductance  Ls  of  the 
oscillating  circuit.  The  condenser  Ci  offers  less  and  less  impedance  to 


EJ05-B E 


Fig.  1. 

the  higher  harmonics,  and  the  latter  will  pass  down  through  Ci  to  the 
filament  terminal.  As  the  first  harmonic  is  always  appreciable,  it  was 
specially  filtered  out  of  the  oscillating  circuit  LSCS,  by  means  of  the  anti- 
resonant  circuit  LzCz.  The  inductance  L2  =  .52  M.H.,  of  course  offered 
some  impedance  to  the  fundamental  frequency,  but  that  was  negligble 
compared  with  the  impedance  that  L\C\  offered  to  the  fundamental. 
These  filters  thus  helped  to  produce  a  pure  sine-wave  oscillation  of  the 
same  frequency  as  the  fundamental  in  the  circuit  LSCS.  The  frequency 
used  throughout  was  200,000  cycles  per  sec.,  or  a  wave-length  of  1,500 
meters.  This  frequency  was  high  enough  that  it  could  be  tuned  very 
sharply,  and  yet  not  so  high  that  the  internal  capacities  of  the  tubes 
would  be  of  any  importance.  Ballantine1  has  worked  out  expressions 
for  the  input  impedence  of  tubes  under  various  conditions,  and  applying 
his  formulae  to  the  W.  E.  205  B  tube  at  this  frequency  and  under  the 
experimental  conditions  which  will  be  described  below,  the  input  impe- 
dance was  found  of  the  order  of  100,000  ohms. 

The  inductance  Z,4  was  loosely  coupled  to  L3  and  connected  by  means 
of  a  twisted  pair  with  L5,  which  in  turn  was  loo  sely  coupled  to  L6.  These 
latter  coils  were  placed  about  seven  meters  away  from  the  oscillating 
circuit,  in  order  that  they  might  not  pick  up  any  of  the  harmonics.  The 

1  PHYS.  REV.,  15,  p.  409-420,  1920. 


NoT6XIX']  THREE-ELECTRODE    VACUUM   TUBES.  64! 

loosening  of  the  couplers  already  described  resulted  in  maintenance  of 
the  sine  e.m.f.  The  combination  of  condensers  C±,  C&,  C&  and  Cy  and  the 
inductance  L6  is  tuned  for  the  fundamental  frequency.  The  arrange- 
ment of  these  condensers  is  what  is  known  as  a  potential  divider  and  has 
been  described  by  Hulbert  and  Breit.1  The  object  is  to  take  a  portion 
of  the  alternating  e.m.f.  across  L6  and  impress  it  on  the  grid  of  a  tube. 
When  the  thermocouple  is  in  the  dotted  position  the  current  passing 
through  C\  is  measured,  but  the  current  passing  through  C5  and  Ce  can 
easily  be  calculated  when  the  values  of  the  different  capacities  are  known. 
The  object  of  measuring  the  current  in  C4  is  that  for  small  values  of  input 
potentials,  the  currents  passing  through  C5  and  C6  would  be  too  small  to 
be  recorded  by  a  low  resistance  thermo-couple.  If  I  represents  the  am- 
plitude of  the  alternating  current  passing  through  C$  and  Ce  then  the 
resulting  input  e.m.f.  is  1/2  IT  (Co  +  Ce)  /,  where  /  is  the  frequency  of  the 
wave.  For  an  e.m.f.  of  over  one  volt,  the  current  /  could  be  measured 
directly  by  the  low  resistance  thermo-couple. 

The  resistance  R%  is  used  to  provide  a  leak  for  any  charge  that  may 
accumulate  on  the  grid,  and  allow  it  to  flow  to  earth.  Its  resistance 
must  be  comparable  to  the  input  impedance  of  the  tube.  By  means  of 
potentiometer  R-A  the  p  otential  on  the  grid  could  be  varied  as  desired. 

The  upper  tube  to  the  right  is  the  harmonic  producer.  By  means  of 
the  condenser  potential  divider  a  known  value  of  input  e.m.f.  was  im- 
pressed between  the  grid  and  filament  and  then  according  to  equations 
(4),  for  a  resistance  EF  in  the  plate  circuit,  and  (9)  for  an  inductance 
EF,  a  plate  current  will  result  which  is  capable  of  being  represented 
as  a  series  of  harmonics.  In  order  to  get  the  results  predicted  in  equa- 
tion (4),  EF  must  be  a  pure  resistance.  A  straight  wire  immediately 
suggests  itself  as  a  resistance  which  would  possess  a  minimum  inductance 
and  capacity.  However,  in  order  to  obtain  a  resistance  of  the  order  of 
3,000  ohms,  so  much  wire  would  be  needed,  that  inductive  and  capaci- 
tive  effects  would  become  appreciable.  Then  again,  it  is  a  well-known 
fact  the  conductivity  of  a  wire  diminishes  with  the  frequency  owing  to 
the  skin  effect,  and  consequently  the  exact  value  of  the  resistance  at  any 
particular  frequency  is  not  easily  determined.  A  resistance  suitable  for 
high-frequency  work  should  have  a  negligible  skin  effect,  as  well  as  hav- 
ing negligible  inductance  and  capacity.  On  the  suggestion  of  Professor 
W.  F.  G.  Swann,  the  author  tried  out  some  platinized  quartz  fibers  im- 
mersed in  acid-free  paraffin  oil,  and  found  that  they  would  carry  currents 
up  to  at  least  60  milliamperes.  From  the  formula  for  change  in  resist- 
ance with  frequency,2  it  can  be  shown  that  using  fibers  about  .01  mm. 

1  PHYS.  REV.,  4,  p.  278,  1920. 

2  J.  A.  Fleming,  Wireless  Telegraphy,  p.  97. 


642  JOHN   G.    FRAYNE.  [SECOND 

LoKRIES . 

in  diameter,  the  skin  effect  can  be  neglected.  As  the  fibers  used  had  a 
resistance  of  about  100  ohms  per  cm.  only  a  short  length  of  circuit  was 
needed,  thus  reducing  the  inductance  and  capacity.  The  inductance 
EFwas  wound  with  No.  16,  D.C.C.  copper  wire,  and  the  windings  were 
spaced  about  I  mm.  apart.  The  resistance  of  the  coil  was  10  ohms, 
whereas  the  reactance  was  2,700  ohms. 

In  order  to  detect  the  various  harmonics,  a  fraction  of  the  e.m.f.  along 
EF  was  impressed  on  the  grid  of  a  W.  E.  Co.  V  tube,  this  impressed  e.m.f. 
being  always  less  than  I  volt.  A  loo-ohm  slide  wire  of  IAIA  wire  was 
used  for  the  variable  portion  of  EF.  It  can  be  seen  from  equation  (8) 
that  in  order  to  obtain  pure  amplification  without  the  introduction  of 
harmonics  whose  amplitudes  are  appreciable  compared  to  that  of  the 
fundamental,  the  input  e.m.f.  must  be  small  (less  than  one  volt),  and 
the  value  of  the  external  resistance  must  be  high.  The  resistance  of  an 
anti-resonant  circuit  is  R  -f-  (L2o>2)/.R,  where  Leo  is  the  inductive  reactance. 
Since  R,  the  ohmic  resistance,  is  negligble,  at  radio  frequencies,  in  com- 
parison with  (LV)/jR,  the  latter  term  may  be  taken  as  the  value  of  the  re- 
sistance of  an  anti-resonant  circuit.  Therefore  for  any  given  co,  L  should 
be  made  as  large,  and  R  as  small  as  possible.  Now  in  the  plate  circuit 
of  the  tube  which  is  used  to  separate  out  the  harmonics,  a  series  of  anti- 
resonant  circuits  are  placed.  The  first  one  is  tuned  for  the  fundamental, 
the  second  for  the  first  harmonic,  and  so  on.  A  vacuum  thermocouple 
is  placed  in  each  circuit  on  the  capacity  side.  This  is  done  so  that  the 
D.C.  plate  current  will  not  affect  it.  In  order  to  keep  the  ohmic  resist- 
ance low,  thermocouples  with  heater  resistances  of  from  0.5  ohm  to  5 
ohms  were  used,  the  higher  resistance  thermo-couples  being  used  to  meas- 
ure the  weaker  amplitudes  of  the  higher  harmonics. 

The  effective  resistance  of  these  circuits  are  as  follows:  fundamental, 
120,000  ohms;  first  harmonic,  183,000  ohms;  second  harmonic,  95,000 
ohms;  third  harmonic,  175,000  ohms. 

Arrangements  (not  shown  in  Fig.  i)  were  also  made  for  measuring 
higher  harmonics  than  these,  by  changing  the  inductance  Ln,  and  by 
retuning  C\\.  Each  thermo-couple  could  be  connected  successively 
to  a  Leeds  and  Northrup  galvanometer,  and  the  deflection  of  the  latter 
indicated  the  root-mean  square  value  of  the  alternating  current  passing 
through  the  heater.  Previous  to  placing  the  thermo-couples  in  the 
circuits,  they  were  calibrated  using  alternating  current  (60  cycles).  It 
will  be  seen  that  the  harmonic  analyzer  is  essentially  a  voltage  amplifier, 
picking  out  each  frequency  in  the  producer  and  magnifying  its  voltage. 
For  this  reason  a  tube  with  a  large  voltage  amplification  constant  was 
chosen,  in  fact  the  value  of  n  as  given  by  equation  (4)  was  26. 


No"6XIX']  THREE-ELECTRODE    VACUUM   TUBES.  643 

Let  1  1  =  the  maximum  current  in  the  fundamental  circuit. 

Let  RI  =  the  effective  resistance. 

Let  R0  =  the  internal  output  resistance  of  the  V  tube. 

Therefore  R\Ii  =  e.m.f.  across  L8. 

Let  e  =  e.m.f.  across  FG. 

Let  M1  =  actual  voltage  amplification  factor. 

™,       ,  ijRi  26  X  120,000 

Therefore  /x1  =  -       —  =  —  —  =  20.8. 


R0+  RI      29,250  +  120,000 

Rill         120,000  X 


.p,          r 

Therefore  e 


Let  r     =  resistance  of  FG. 

Let  i      =  amplitude    of    current    of    fundamental    frequency 
passing  through  FG. 

™,       r  ,  .       120,000.  /i  ,     N 

Inereiore  n         =  e  and  *  =  • —  -  .  (15) 

20.8  X  r 

Thus  knowing  /i  from  the  galvanometer  deflection,  and  r  from  the 
Wheatstone  Bridge,  the  value  of  i  can  be  determined.  For  the  case 
worked  out  above  i  represents  the  amplitude  of  the  fundamental  fre- 
quency produced  by  a  pure  sine  wave  impressed  on  the  grid  of  a  tube 
having  a  pure  resistance  load  in  the  plate  circuit.  Similarly,  by  measur- 
ing the  currents  in  the  other  tuned  circuits  we  can  work  back  to  the 
equivalent  current  in  the  harmonic  producer. 

When  the  resistance  EF  is  replaced  by  an  inductance,  a  portion  FG 
of  the  inductance  is  used  to  obtain  the  input  on  the  grid  of  the  analyzer. 
In  this  case  it  will  be  noted  that  EF  offers  twice  as  much  impedance  to 
the  first  harmonic,  three  times  as  much  to  the  second  harmonic,  and  so 
on.  This  makes  it  possible  to  measure  weaker  harmonics  than  in  the 
case  of  the  resistance.  The  inductance  of  GF  in  this  experiment  was 
0.0587  milli-henry,  whereas  the  whole  inductance  of  EF  was  2.14  milli- 
henries. The  input  impedance  of  the  analyzer  to  which  EF  was  at- 
tached was  of  the  order  of  50,000  ohms  at  200,000  cycles  per  sec.,  whe  reas 
the  impedance  of  .0587  henry  is  only  74  ohms.  This  showst  hat  the 
impedance  of  the  analyzer  was  practically  short-circuited  by  the  coil 
FG,  and  consequently  did  not  affect  the  nature  of  the  external  circu  it  of 
the  producer.  For  measurement  of  large  output  current  values,  the 
value  of  FG  was  reduced  to  0.04  M.H.  To  obtain,  say  the  amplitude 
of  the  fundament  current  with  the  inductance,  we  have  an  equati  on 


644 

analogous  to  (15), 


JOHN   G.   FRAYNE. 


[SECOND 
[SERIES. 


1   = 


I2O.OOO  / 


20.8  (/w)   ' 

where  I  is  the  inductance  of  FG  and  co  =  2  TT  X  the  frequency. 

EXPERIMENTAL  RESULTS. 

The  following  constants  for  the  2O5-J5  tube  were  determined  from  its 
static  characteristic.  A  —  .554  X  io~6,  e  =  7.5  volts,  /*  =  6.7.  For  Eb 
=  260  volts,  Ec  =  —  7.5  volts,  Ro  =  1/2  A  (Eb  +  nEc  +  «)*=  3,570  ohms. 


200  volt* 
<o.  t    CC--T*    "     R- 2700  Oh 


Resistance  in  Plate  Circuit. — When  a  resistance  of  2,700  ohms  was 
placed  in  series  with  the  plate,  and  the  value  of  Eh  reduced  to  200  volts, 
the  plate  current  was  21.5  milliamperes.  Using  these  values  of  poten- 
tial and  resistance  the  curves  shown  in  Fig.  2  were  obtained.  The  ampli- 
tude of  the  input  e.m.f.  was  varied  from  o  up  to  50  volts.  The  latter 
maximum  loaded  the  tube  rather  heavily,  and  the  larger  currents  were 
maintained  just  long  enough  to  obtain  the  necessary  readings. 

Keeping  the  e.m.f.  of  the  input  at  15  volts,  the  actual  plate  potential 
at  200  volts,  and  varying  the  static  grid  potential  the  curves  in  Fig.  3 
were  obtained,  for  ranges  of  Ec  between  —  30  and  +12  volts. 

With  a  static  voltage  of  —  7.5  on  the  grid  and  the  alternating  e.m.f. 
kept  at  15  volts,  the  variation  of  the  harmonics  with  plate  voltage  was 
determined,  as  in  Fig.  4. 


VOL.  XIX.1 
No.  6. 


THREE-ELECTRODE    VACUUM    TUBES. 


645 


-  Fundamental 
B    I*  Harmonic 
C 


Volts  -  Et 


Ra  4 


646 


JOHN   G.   FRAYNE. 


[SECOND 

[SERIES. 


A -Fundamental. 
3-  ft  Harmonic 


E-  zoo  volts 


K  *    2700  Ohms. 


Fig.  5  represents  the  wave-shape  of  the  plate  current  obtained  when  a 
pure  sine  wave  e.m.f.  of  15  volts  is  impressed  on  the  grid,  the  plate  and 
grid  potentials  being  the  same  as  stated  above.  The  fundamental  and 
first  harmonic  are  the  only  components  included  in  the  wave-shape.  The 


VOL.  XIX.l 
No.  6. 


THREE-ELECTRODE    VACUUM   TUBES. 


647 


amplitudes  of  the  other  harmonics  are  too  small  to  be  shown  on  the  same 
scale.  Fig.  5  also  shows  the  dynamic  characteristic  for  this  case,  where 
the  wave-shape  thus  obtained  is  plotted  against  the  alternating  input 
voltage. 

In  Fig.  6  the  variation  of  the  harmonics  with  the  value  of  the  external 
resistance  is  shown. 

Inductance  in  Plate  Circuit. — Exactly  similar  experimental  procedure 
was  undertaken  for  the  inductances.  A  higher  plate  voltage  and  plate 
current  was  used  here,  since  there  were  no  delicate  platinized  quartz  fibers 
to  be  dealt  with.  For  this  case 


=  250  volts,  EC  =  —  10  volts,  ^0  = 


2A  (Eb 


=  3,800  ohms 


Theoretical 
A      Fundamental 
Harmonic 


The  reactive  load  in  the  plate  circuit  was  2,700  ohms.  Fig.  7  shows 
the  variation  of  the  harmonics  with  the  alternating  e.m.f.  on  the  grid, 
under  the  conditions  given  above. 

In  Fig.  8  is  shown  the  variation  of  the  harmonics  with  the  value  of 
the  static  grid  potential,  the  alternating  input  e.m.f.  being  constant  at 
20  volts,  and  the  plate  voltage  being  250  volts.  Fig.  9  shows  how  the 
variation  of  the  static  plate  voltage  affects  the  values  of  the  harmonics. 

The  curves  of  Fig.  10  give  the  variation  of  the  harmonics  with  the 
magnitude  of  the  inductance  in  the  plate  circuit. 

Fig.  ii  represents  the  wave  shape,  using  only  the  values  of  the  fun- 
damental and  first  harmonic. 


648 


JOHN   G.   FRAYNE. 


[SECOND 
[SERIES  . 


A-  fundamental 

B-  KHormonic 

C-2*    " 

0-3*    " 

E-4*    » 

£.750 

V-15     -    .L-2-/4MH 


I-nJuctanct  -Z/4AW 

s  s  s    A -Fundamental 
5  -v         B  -  1^  HaTTnoni'c 
fs?    C 

t)    3" 
E    4» 


VOL.  XIX.1 
No.  6. 


THREE-ELECTRODE    VACUUM   TUBES. 


649 


A- 

B-K  Harmonic. 

C-pf   »  F)(p- 


Thcor. 


C  r.         .  ' "  *•» 

Fig.  ia 

Fig.  II  also  represents  the  dynamic  characteristics  for  this  case,  the 
area  of  this  loop  being  proportional  to  the  amount  of  energy  delivered. 


A-  Fundamental 
C-  |tf  Hahmonic 
B-  Wave-Sha^e 


D-  Dfnam/c 
J 
ISO  Volts 

EC  •  -10       ' 
L  '  X-  lit   77)  h 

•t"  *  20     VbltS. 


Fig.  11. 

DISCUSSION  OF  RESULTS. 

The  theoretical  curves  as  shown  in  the  various  figures,  are  plotted  from 
the  values  of  the  coefficients  given  by  equations  (3)  and  (8) .  In  the  case 
of  the  resistance  the  experimental  fundamental  values  check  up  very 
closely  with  the  theoretical  values  up  to  an  input  voltage  of  fifteen  volts. 


650  JOHN   G.    FRAYNE. 

Beyond  that  voltage  equation  (3)  no  longer  accurately  represents  con- 
ditions. It  will  be  noticed  that  the  maximum  positive  potential  to  which 
the  grid  is  raised  in  this  operation  is  7.5  volts.  Fig.  5  shows  that  at  this 
voltage  the  static  characteristic  begins  to  flatten  out,  due  to  the  passage 
of  electrons  to  the  grid  instead  of  to  the  plate.  For  voltages  higher  than 
fifteen  the  theoretical  first  and  third  harmonics  fall  below  the  experi- 
mental values,  whereas  the  second  harmonic  is  greater  than  the  experi- 
mental values  would  indicate.  This  would  seem  to  be  due  to  the  flat- 
tening out  of  the  wave  at  the  upper  end  of  the  characteristic.  In  the 
case  of  the  inductance  good  agreement  between  theoretical  and  experi- 
mental values  were  found  for  input  voltages  as  high  as  20  volts.  This 
is  due  to  the  fact  in  this  case  that  the  operation  was  carried  out  over  the 
25o-volt  static  characteristic,  and  it  will  be  seen  from  Fig.  n  that  this 
curve  does  not  begin  to  flatten  out  until  a  positive  grid  voltage  of  ten 
volts  is  reached.  This  point  of  operation  coincides  with  the  maximum 
positive  voltage  to  which  the  grid  was  raised  with  an  input  e.m.f.  of 
20  volts,  Ec  being  —  10  volts. 

The  curves  of  Figs.  3  and  8  show,  in  an  emphatic  manner,  that  the 
further  we  move  towards  the  straighter  portion  of  the  static  character- 
istic, the  greater  the  fundamental  becomes  while  the  other  harmonics 
continue  to  diminish.  At  —  40  volts  the  higher  harmonics  were  still 
very  much  in  evidence  although  the  fundamental  was  rapidly  approach- 
ing zero.  The  second  harmonic  in  the  resistance  curves  and  the  third 
in  the  inductance  curves  show  rather  peculiar  irregularities.  The  sharp 
maxima  and  minima  would  at  first  seem  to  point  to  some  sort  of  internal 
resonance  in  the  tube.  It  will  be  noticed,  however,  that  these  are  pro- 
duced by  simply  varying  either  the  grid  or  plate  potentials,  and  are 
probably  due  to  irregularities  in  the  static  characteristic  which  are 
smoothed  out  in  the  ordinary  process  of  plotting.  It  will  be  recalled 
that  in  the  case  of  the  resistance  the  coefficient  of  the  second  harmonic 
is  principally  determined  by  the  value  of  the  third  derivative  of  the  cur- 
rent function.  A  small  unnoticeable  irregularity  in  the  static  charac- 
teristic might  produce  a  large  variation  in  the  third  derivative,  and  this 
effect  is  magnified  by  this  method  of  analysis.  In  the  case  of  the  induc- 
tance the  coefficients  cannot  be  expressed  as  simple  derivatives,  but  some 
such  explanation  as  that  given  above  will  probably  hold  good  here  also. 

The  curves  of  Figs.  4  and  9  show  that  the  greater  the  plate  potential 
the  greater  the  fundamental  becomes  while  the  harmonics  continue  to 
diminish.  Beyond  a  certain  plate  potential,  about  300  volts,  the  funda- 
mental becomes  nearly  constant  in  value.  The  curves  which  exhibited 
the  irregularities  in  the  grid-variation  series,  also  exhibit  similar  irregu- 


NoL6XIX']  THREE-ELECTRODE   VACUUM    TUBES.  65! 

larities  here.  Further  they  occur  at  the  same  value  of  (Et,  -f-  /j.Ec) 
showing  that  the  effect  is  independent  of  whether  the  grid  or  plate  poten- 
tial is  increased  provided  the  sum  as  given  is  the  same. 

The  curves  of  Figs.  6  and  10  show  how  the  harmonics  depend  on  the 
external  impedance.  Using  this  method  of  analysis  it  was  impossible 
to  use  an  impedance  much  less  than  500  ohms.  Even  at  this  value,  the 
input  impedance  of  the  analyzing  tube  cannot  be  neglected.  Hence, 
beyond  the  first  harmonic  the  experimental  and  theoretical  curves  do 
not  agree  very  closely  in  this  region.  Although  the  experimental  curves 
show  signs  of  flattening  out  at  this  low  impedance,  they  should  have  been 
rapidly  approaching  zero.  For  zero  resistance  or  inductance,  the  funda- 
mental and  first  harmonic  have  the  values  given  by  equation  (20),  and 
all  the  other  harmonics  are  zero,  which  is  also  in  accordance  with  the 
same  equation. 

If  all  the  harmonics  were  neglected  the  wave-shape  for  the  resistance 
would  be  a  pure  sine  wave  in  phase  with  the  impressed  e.m.f.  These 
two,  when  compounded,  would  give  a  straight-line  dynamic  character- 
istic. It  may  be  seen  that  with  a  sufficiently  high  resistance  in  the  plate 
circuit  this  condition  may  be  nearly  reached.  However,  if  the  higher 
harmonic  were  also  taken  into  consideration,  the  dynamic  characteristic 
would  be  no  longer  linear  but  would  have  a  curvature,  which  would  be 
much  less  than  that  of  the  static  curve.  With  the  inductance,  if  all  but 
the  fundamental  had  been  neglected  the  current  would  have  been  a  pure 
sine  wave  lagging  behind  the  impressed  e.m.f.  by  an  angle  6  =  tan~llp/R. 
The  dynamic  characteristic  under  those  conditions  would  have  been  an 
ellipse.  Addition  of  the  other  harmonic  components  tends  to  flatten 
out  the  sine  wave,  and  consequently  distort  the  purely  elliptical  charac- 
teristic. 

In  calculating  the  theoretical  values  of  the  coefficients  of  the  various 
harmonics,  up  to  a  range  of  15  or  20  volts,  the  coefficients  of  the  funda- 
mental were  practically  proportional  to  the  first  power  of  the  applied 
e.m.f.;  the  first  harmonic  was  proportional  to  the  second  power;  and  so 
on.  It  was  only  for  values  of  the  input  voltage  beyond  20  volts  that  the 
more  complicated  expressions  for  the  coefficients  had  to  be  evaluated. 

In  conclusion,  the  writer  wishes  to  express  his  very  sincere  thanks  to 
the  Western  Electric  Company,  Inc.,  of  New  York,  for  their  kindness  in 
loaning  necessary  tubes  and  vacuum  thermo-couples,  and  also  to  Profes- 
sor W.  F.  G.  Swann  of  this  department  for  his  many  helpful  suggestions 
and  criticisms  and  his  invaluable  encouragement  at  all  times. 

DEPARTMENT  OF  PHYSICS, 

UNIVERSITY  OF  MINNESOTA, 
November  n,   1921. 


IHX8B     ^s^TyffED^  ^g 

**yz.         ^^^^^ 


OVERDUE 


Gay  lord  Bros. 

Makers 
Syracuse,  N.  Y. 

PAT.  JAM  21,  1908 


487359 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


